Nexus Number

A Figurate Number built up of the nexus of cells less than $n$ steps away from a given cell. In $k$-D, the $(n+1)$th nexus number is given by

$$N_{n+1}(k)=\sum_{i=0}^{k} {k\choose i}n^i,$$

where ${n\choose n}$ is a Binomial Coefficient. The first few $k$-dimensional nexus numbers are given in the table below.

$k$	$N_{n+1}$	name
0	1	unit
1	$1+2n$	Odd Number
2	$1+3n+3n^2$	Hex Number
3	$1+4n+6n^2+4n^3$	Rhombic Dodecahedral Number

See also Hex Number, Odd Number, Rhombic Dodecahedral Number

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 53-54, 1996.